Subthreshold dynamics of the neural membrane potential driven by stochastic synaptic input

Subthreshold dynamics of the neural membrane potential

driven by stochastic synaptic input

Ulrich Hillenbrand∗

Institute of Robotics and MechatronicsGerman Aerospace Center

Oberpfaffenhofen, 82234 Wessling, Germany

In the cerebral cortex, neurons are subject to a continuous bombardment of synaptic inputsoriginating from the network’s background activity. This leads to ongoing, mostly subthresholdmembrane dynamics that depends on the statistics of the background activity and of the synapsesmade on a neuron. Subthreshold membrane polarization is, in turn, a potent modulator of neuralresponses. The present paper analyzes the subthreshold dynamics of the neural membrane po-tential driven by synaptic inputs of stationary statistics. Synaptic inputs are considered in linearinteraction. The analysis identifies regimes of input statistics which give rise to stationary, fluctu-ating, oscillatory, and unstable dynamics. In particular, I show that (i) mere noise inputs can drivethe membrane potential into sustained, quasiperiodic oscillations (noise-driven oscillations), in theabsence of a stimulus-derived, intraneural, or network pacemaker; (ii) adding hyperpolarizing todepolarizing synaptic input can increase neural activity (hyperpolarization-induced activity), in theabsence of hyperpolarization-activated currents.

PACS numbers: 87.19.La, 87.10.+e, 02.50.-r.

Keywords: neural networks, cerebral cortex, membrane potential, synapses, stochastic process.

Published as Physical Review E 66, 021909 (2002).

I. INTRODUCTION

Cortical pyramidal cells fire action potentials at an average spontaneous rate of about 10 spikes/s in waking animals[27, 28]. At such a low spike rate, it is clear that most cortical neurons spend a significant amount of time with theirmembrane potential well below the threshold for spike activation. On the other hand, a cortical pyramidal cell receivesroughly 10000 synapses [8], mostly from other cortical neurons. Since individual postsynaptic events cause transientincreases in membrane conductance, it follows that the dynamics of membrane potentials is largely controlled bysubthreshold stimulation from the continuous network activity. Subthreshold membrane polarization is, in turn, apotent modulator of stimulus-driven spike activity [3, 23].

In this paper, I analyze the subthreshold dynamics of the membrane potential driven by stochastic synaptic activityof general stationary statistics. Such conditions are given in neurons that do not respond to an external stimulus,but are exposed to the network’s spontaneous or stimulus-driven background activity. The generation of postsynapticpotentials (PSPs) and their propagation along the dendrites of a neuron are modeled in a rather simple way toallow for a thorough analytical treatment. Accordingly, the focus is on generic patterns of behavior rather than onquantitative results. Some of the conclusions are discussed in relation to the experimental literature.

II. MODELING SYNAPTIC RESPONSES

The potential V across a local patch of passive membrane is described by

d

dtV = − 1

τmV +

1

τm gmI , (1)

where τm and gm are the passive membrane time constant and leak conductance, respectively, and I is the currentpassed along the dendrites from other parts of the cell. The membrane’s resting potential is set to zero. After asynaptic input has been received on the considered patch of membrane, the potential obeys

d

dtV = − 1

τmV +

1

τm gmI +

gsτm gm

(Vs − V ) , (2)

∗[email protected]

2

where Vs and gs are the synaptic reversal potential and conductance, respectively. Let V0(t) and Vin(t) be solutions toEqs. (1) and (2), respectively, with V0(0) = Vin(0) = V (0). Synaptic ion channels are open for a brief period δs ¿ τm[31]. At time t = δs, when synaptic channels close, the deflection of the membrane potential due to the synaptic inputis

Vin(δs)− V0(δs) =δsτm

gsgm

[Vs − V (0)] +O

[(δsτm

)2]

. (3)

This deflection propagates along the cell’s dendrites. Far away from its point of origin, I model the synaptic responseas a PSP. In a passive cable, the rise time and amplitude of a PSP depend on the time course of the synaptic current,and the relative locations of the synapse and the point on the membrane at which the PSP is observed; the decay-time constant approaches τm for long times [24–26, 30]. However, computer-simulation studies involving realistic cellmorphologies [7, 19] and voltage-dependent dendritic conductances [9] have revealed that PSPs in real neurons maybe less variable than suggested by a cylindrical passive-cable model. A coarse but, for the present analysis, sufficientapproximation to a PSP is given by the impulse response of a second-order low-pass filter,

Λ(γ, Vs, t0; t) := γ [Vs − V (t0)]t− t0τ

exp

(

1− t− t0τ

)

Θ(t− t0) , (4)

with the unit-step function

Θ(t) :=

{0 for t ≤ 0,1 for t > 0.

(5)

The PSP’s amplitude is γ [Vs − V (t0)], with the factor

γ := aδsτm

gsgm

> 0 . (6)

Thus, the PSP is initiated at time t0, has a rise time and decay-time constant τ , is attenuated or amplified by a factora [cf. Eq. (3)], and is assumed to propagate instantaneously. It qualitatively captures the basic properties of real PSPsof having a finite rise time and an exponential decay phase. It is chosen here for its convenience for analysis.

Postsynaptic conductance changes are very local compared to the extended dendritic trees on which synapses makecontacts. It is therefore a reasonable approximation to treat them as noninteracting. The total membrane potentialunder synaptic control is hence given by the sum

V (t) =

∞∑

i=1

Λ(γi, si, ti; t) (7)

for the whole cell. Here t1 ≤ t2 ≤ . . . are the times of synaptic input received by a neuron; γi and si are theamplitude-related factor defined in Eq. (6) and the reversal potential of the ith synaptic input, respectively. In Sec.III E, I will address effects of delays in the propagation of PSPs.

III. ANALYSIS AND RESULTS

Upon inspection of Eqs. (4) and (7), it is clear that there is an equivalence relation between the statistics of the γiand of the pairs (si, ti). Higher values of γi have the same effect on the dynamics of V (t) as shorter intervals ti+1− tibetween successive stimuli with si = si+1. In order to simplify the analysis, without limiting the dynamic repertoireof V (t), it is preferable to restrict to one value γ ≡ γi. In this section, I shall thus derive analytical results on thedynamics

V (t) = γ

∞∑

i=1

[si − V (ti)]t− tiτ

exp

(

1− t− tiτ

)

Θ(t− ti) . (8)

Moreover, the results will be illustrated by computer simulations where appropriate.Arguably, the “obvious” approach to the problem is to specify the distribution functions for the point process that

models the times ti of stimulus events and write down integral equations for the moments of V (t). However, weshall take a different approach. We will start by casting the dynamics in the form of a Markov chain. There aretwo significant advantages proceeding this way. First, it will allow us to go quite far with the analysis without beingspecific about the stimulus process. Only at some later point will it be profitable to specify the statistics of stimulustimes. Second, making use of the Markov property, we will gain insight not only into the dynamics of moments of themembrane potential, but also into the temporal pattern of individual trajectories V (t).

3

A. Markov formulation of the dynamics of the membrane potential

Introducing the notation

xj := V (tj) = γ

j−1∑

i=1

[si − V (ti)]tj − tiτ

exp

(

1− tj − tiτ

)

, (9)

yj := γ

j−1∑

i=1

[si − V (ti)] exp

(

− tj − tiτ

)

, (10)

rj := tj+1 − tj , (11)

we can reformulate the dynamics of Eq. (8) for the discrete times t = tj as an iteration of a combination of twostochastic maps R(r) and S(s),

(xjyj

)

= R(rj−1) ◦ S(sj−1)(xj−1yj−1

)

, x1 = y1 = 0 , (12)

S(s) :(x

y

)

7→(

x

y + γ (s− x)

)

, (13)

R(r) :(x

y

)

7→((

x+ ey rτ

)e−r/τ

ye−r/τ

)

. (14)

The interstimulus times rj and the synaptic reversal potentials sj are stochastic variables, drawn independently fromdensities u(r) on R+ and v(s) on R, respectively. These densities are determined by the neural network activityand the number and types of synapses on the neuron considered. Note that although there may well be statisticaldependences between rj and sj , and (rj , sj) and (rj′ , sj′) (j 6= j′) as sampled at one individual synapse, these do notshow up in the sequences rj and sj for all synaptic inputs to a cortical neuron.

In the present formulation of the dynamics, the synaptic input times tj are, like xj and yj , stimulus-driven stochasticvariables and may be incorporated by extending the system (12) with the equation

tj = tj−1 + rj−1 . (15)

This equation can be solved independently of Eq. (12). In particular,

〈tj〉 = (j − 1) 〈r〉+ t1 . (16)

Here and in the following, we encounter mean values of the types

〈f(s)〉 :=∫ ∞

−∞

ds′ v(s′) f(s′) , 〈f(r)〉 :=∫ ∞

0

dr′ u(r′) f(r′) , (17)

with f being some function on the real numbers for which the integrals are defined.The dynamics (12) is a Markov chain. The transition probability corresponding to S(s) is

pS(x, y|x′, y′) =∫ ∞

−∞

ds v(s) δ(x− x′) δ[y − y′ − γ(s− x′)] , (18)

and the one corresponding to R(r) is

pR(x, y|x′, y′) =∫ ∞

0

dr u(r) δ[

x−(

x′ + ey′r

τ

)

e−r/τ]

δ(

y − y′e−r/τ)

. (19)

Here δ is the Dirac delta function. Let p(x, y) be a joint probability density for x and y. Then

〈xnym〉 :=∫ ∞

−∞

dx′∫ ∞

−∞

dy′ p(x′, y′)x′ny′m , n,m ∈ N , (20)

4

are the moments of x and y. We want to know how the moments change under the action of R(r) ◦ S(s). For theaction of S(s), we get

〈xnym〉S =

∫ ∞

−∞

dx

∫ ∞

−∞

dy

∫ ∞

−∞

dx′∫ ∞

−∞

dy′ pS(x, y|x′, y′) p(x′, y′) xnym

=∑

h,i,j∈Nh+i+j=m

(m

h, i, j

)

(−1)iγh+i⟨sh⟩ ⟨xn+iyj

⟩, (21)

with polynomial coefficients

(m

h, i, j

)

:=m!

h! i! j!, h+ i+ j = m . (22)

The action of R(r) yields

〈xnym〉R =

∫ ∞

−∞

dx

∫ ∞

−∞

dy

∫ ∞

−∞

dx′∫ ∞

−∞

dy′ pR(x, y|x′, y′) p(x′, y′) xnym

=

n∑

k=0

(n

k

)⟨(er

τ

)k

e−(n+m)r/τ⟩⟨xn−kym+k

⟩. (23)

Let pj(x, y) be the joint probability density of x and y at time tj . By combining Eqs. (21) and (23), we can writedown iteration equations for the moments,

〈xnym〉j :=∫ ∞

−∞

dx′∫ ∞

−∞

dy′ pj(x′, y′)x′ny′m . (24)

The iterations can be solved successively for all n and m, starting with the first moments. We shall solve for thefirst two moments, i.e., for 〈x〉j , 〈y〉j , 〈x2〉j , 〈xy〉j , and 〈y2〉j . Note that the ensemble averages (24) are not taken atconstant time t, but rather at a constant number j of synaptic inputs received, irrespective of the time tj of the jthinput. As mentioned above, the times of synaptic inputs are additional random variables obeying Eq. (15).

B. Mean membrane potential

The iteration dynamics of the mean values obtained from Eqs. (21) and (23) is

(〈x〉j〈y〉j

)

=

(a1 − γb1 b1−γa1 a1

)

︸ ︷︷ ︸

=:M1

(〈x〉j−1〈y〉j−1

)

+ γ 〈s〉(b1a1

)

, 〈x〉1 = 〈y〉1 = 0 , (25)

with the stimulus parameters 〈s〉 and

a1 :=⟨e−r/τ

⟩

b1 :=⟨rτ e1−r/τ

⟩

}

∈ (0, 1) . (26)

The dynamics of 〈x〉j and 〈y〉j depend on the eigenvalues of M1, and thus on the stimulus parameters a1 and b1. Theeigenvalues are

λ1/2 := a1 −γb12± 1

2

√

γ2b21 − 4γa1b1 . (27)

For convergence of the dynamics, we require that∣∣λ1/2

∣∣ < 1 ⇐⇒ γb1 < (a1 + 1)2 . (28)

Figure 1 shows the parameter regions of convergence and divergence. In this parameter space, the vicinity of thepoint a1 = 1, b1 = 0 is occupied by high-frequency stimuli, i.e., with short interstimulus times r. A very low networkactivity, on the other hand, lies close to the point a1 = 0, b1 = 0. It turns out that for any input statistics, the mean

5

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Unstable

PSfrag replacements

a1

γb1

FIG. 1: Space of stimulus parameters a1 and b1 that determine the dynamics of the mean membrane potential. The dynamicsconverges for (a1 + 1)2 > γb1. For γb1 < 4a1, the two eigenvalues given by Eq. (27) are complex conjugate. For (a1 + 1)2 >γb1 > 4a1, they are real and negative. The corresponding type of mean dynamics is depicted for these two regimes.

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

1

2

3

4

PSfrag replacements

a1a1

γb1

γ = 0.2

γ = 0.2

γ = 3.7

γ = 3.7

〈x〉∞ / 〈s〉

FIG. 2: Left: Contour plot of the asymptotic mean membrane potential 〈x〉∞. Dashed lines delimit the regions of differentmean dynamics shown in Fig. 1. Assuming Poisson statistics for stimulus times, the stimulus parameters a1 and b1 lie onparabolas, here plotted for γ = 0.2, 0.7, . . . , 3.7. Right: Plot of 〈x〉∞/〈s〉 for Poissonian stimulus times and the same values ofγ as on the left. The curves are interrupted where (a1 + 1)2 < γb1 such that the mean dynamics is divergent; cf. Fig. 1.

of the membrane potential V converges, if the factor γ, controlling PSP amplitudes, is sufficiently small. For γb1 > 4,on the other hand, the mean dynamics will never converge.

From Eq. (25) we obtain the asymptotic values

limt→∞

〈V (t)〉 = 〈x〉∞ =γb1

γb1 + (1− a1)2〈s〉 , (29)

〈y〉∞ =γ(1− a1)a1

γb1 + (1− a1)2〈s〉 (30)

for the regime of convergence. A contour plot of 〈x〉∞ as a function of a1 and γb1 is shown in the left graph of Fig.2. For the times tj of synaptic input being consistent with a Poisson process, it is shown in the Appendix that thestimulus parameters a1, b1 lie on a parabola, plotted in the left graph of Fig. 2 for different γ. The ratio 〈x〉∞/〈s〉behaves then as shown in the right graph of the figure. Not surprisingly, the mean membrane potential is pulled closerto the mean synaptic reversal potential 〈s〉 with increasing a1, that is, with increasing stimulus frequency, and withincreasing PSP amplitude γ.

For γb1 < 4a1, the eigenvalues λ1/2 are complex conjugate. In Sec. III C, I will show that only then will the varianceof V (t) converge. As depicted in Fig. 1, in this regime 〈V (t)〉 converges in a damped oscillation. The dynamics is

6

solved straightforwardly. Let L be the matrix that diagonalizes M1, i.e., LM1L−1 is diagonal. Furthermore, let

K :=

(1 1i −i

)

. (31)

Then the matrix KL is real and we can write the powers of M1 as

M j1 = aj1L

−1K−1

(cos(jφ) − sin(jφ)sin(jφ) cos(jφ)

)

KL , with φ := arg(λ1) . (32)

The iteration dynamics (25) is solved by

(〈x〉j〈y〉j

)

=

(〈x〉∞〈y〉∞

)

−M j−11

(〈x〉∞〈y〉∞

)

, (33)

that is, a spiral motion around the attractive focus (〈x〉∞ , 〈y〉∞). Its angular period is, measured in the number ofsynaptic input events,

P =2π

arg(λ1), (34)

and averages in real time to

〈T 〉 = P 〈r〉 . (35)

Thus 〈T 〉 is the mean period of 〈V (t)〉. Moreover, it may be shown easily that P is the period of the covariancefunction,

cov(xj , xj+k) :=⟨(

xj − 〈x〉j)(

xj+k − 〈x〉j+k)⟩

=

∫ ∞

−∞

dx

∫ ∞

−∞

dy

∫ ∞

−∞

dx′∫ ∞

−∞

dy′ pk(x, y|x′, y′) pj(x′, y′)(

x− 〈x〉j+k)(

x′ − 〈x〉j)

, (36)

where

p1(x, y|x′, y′) :=

∫ ∞

−∞

dx

∫ ∞

−∞

dy pR(x, y|x, y) pS(x, y|x′, y′) , (37)

pk(x, y|x′, y′) :=

∫ ∞

−∞

dx

∫ ∞

−∞

dy p1(x, y|x, y) pk−1(x, y|x′, y′) for k > 1.

In particular, the asymptotic covariance function limj→∞ cov(xj , xj+k) alternates between phases of correlation andanticorrelation with period P . In Sec. IIID, I will show that under certain stimulus conditions these oscillations ofthe membrane potential never die out for individual realizations of the stochastic process. The damping of the meanoscillation is then due to a loss of phase coherence with time.

For Poissonian stimulus times tj , the mean oscillation period is given by

⟨T

τ

⟩

=

2π⟨rτ

⟩/

arctan

[√eγ〈r/τ〉[(4−eγ)〈r/τ〉+4]

(2−eγ)〈r/τ〉+2

]

for (2− eγ)⟨rτ

⟩+ 2 > 0,

2π⟨rτ

⟩/{

π + arctan

[√eγ〈r/τ〉[(4−eγ)〈r/τ〉+4]

(2−eγ)〈r/τ〉+2

]}

elsewhere;(38)

cf. the Appendix. Figure 3 shows plots of 〈T/τ〉 for different γ, both as a function of 〈r/τ〉 and a1. For 〈r/τ〉 >4/(eγ − 4) or, equivalently, a1 < 1− 4/(eγ), the stimulus enters the regime where λ1/2 are real and negative, and themean period ends up on the curve

⟨T

τ

⟩

= 2⟨ r

τ

⟩

= 2

(1

a1− 1

)

, (39)

plotted with the dashed lines in Fig. 3. For 〈r/τ〉 → 0 or, equivalently, a1 → 1, we find that 〈T/τ〉 approaches zero.In particular, 〈T 〉 can be much shorter than the rise time τ of PSPs.

7

0.2 0.4 0.6 0.8 1

2.5

5

7.5

10

12.5

15

17.5

20

1 2 3 4 5

2.5

5

7.5

10

12.5

15

17.5

20

PSfrag replacements

a1 〈r/τ〉

γ = 0.2 γ = 0.2

γ = 3.7 γ = 3.7

〈T/τ〉〈T/τ〉

FIG. 3: Mean oscillation period 〈T 〉 of the membrane potential in units of the rise time τ of PSPs [cf. Eq. (4)], plotted asa function of the stimulus parameters a1 (left) and 〈r/τ〉 (right). For the curves we assume Poisson statistics for stimulustimes and γ = 0.2, 0.7, . . . , 3.7. The mean oscillation period lies on the dashed curves for a1 < 1 − 4/(eγ) or, equivalently,〈r/τ〉 > 4/(eγ − 4).

C. Variance of the membrane potential

To estimate whether the trajectories V (t) stay bounded when their mean converges to a finite value, we have tocheck whether their variance converges as well. We will now analyze the dynamic map for the second moments of xand y defined in Sec. IIIA. From Eqs. (21) and (23), we obtain

⟨x2⟩

j

〈xy〉j⟨y2⟩

j

=

a2 − γb2 + γ2c2 b2 − 2γc2 c2

−γa2 + γ2

2 b2 a2 − γb212b2

γ2a2 −2γa2 a2

︸ ︷︷ ︸

=:M2

⟨x2⟩

j−1

〈xy〉j−1⟨y2⟩

j−1

+

uj−1vj−1wj−1

, (40)

⟨x2⟩

1= 〈xy〉1 =

⟨y2⟩

1= 0 ,

with the stimulus parameters

a2 :=⟨e−2r/τ

⟩

b2 :=⟨2rτ e1−2r/τ

⟩

c2 :=⟨(

rτ

)2e2−2r/τ

⟩

∈ (0, 1) , (41)

and

uj :=(γb2 − 2γ2c2

)〈s〉 〈x〉j + 2γc2 〈s〉 〈y〉j + γ2c2

⟨s2⟩, (42)

vj :=(γa2 − γ2b2

)〈s〉 〈x〉j + γb2 〈s〉 〈y〉j +

1

2γ2b2⟨s2⟩, (43)

wj := −2γ2a2 〈s〉 〈x〉j + 2γa2 〈s〉 〈y〉j + γ2a2⟨s2⟩. (44)

The 〈x〉j and 〈y〉j converge to the values given in Eqs. (29) and (30) such that (uj , vj , wj) will become constant. Tocheck convergence of the second moments, it is thus necessary and sufficient to consider the eigenvalues of M2. Theseare the roots of the characteristic polynomial

ν3 −(3a2 − 2γb2 + γ2c2

)ν2 +

(

3a22 − γ2a2c2 − 2γa2b2 +1

2γ2b22

)

ν − a32 = 0 , (45)

and are rather lengthy expressions which need not be spelled out here. Depending on the stimulus parameters a2, b2,and c2, there are one real and two complex conjugate eigenvalues, or three real eigenvalues. Let ν1 be the eigenvaluethat is always real and ν2/3 the other two that may be complex conjugate or real. Stimulus parameters a2, b2, c2 thatyield a convergent second moment of V (t) are those that obey the constraints

|ν1| =: f1(a2, γb2, γ2c2) < 1 , max (|ν2| , |ν3|) =: f2(a2, γb2, γ

2c2) < 1 , (46)

8

with continuous functions f1 and f2. The two surfaces defined by

f1(a2, γb2, γ2c2) = 1 , f2(a2, γb2, γ

2c2) = 1 (47)

are shown in Fig. 4. Since convergence is obviously ensured for γ = 0, which yields xj ≡ yj ≡ 0 [cf. Eq. (12)], theparameter region that results in convergence of the second moments is the space between the two surfaces that includesthe axis (a2, γb2, γ

2c2) = (a2, 0, 0), a2 ∈ (0, 1). In the region beyond the intersection of the surfaces, i.e., for roughlyγ2c2 > 9, there are no combinations of parameters that yield convergent second moments.

For Poisson statistics of the stimulus times tj , the stimulus parameters a2, b2, c2 lie on the curves plotted in Fig.4 for different values of γ; cf. the Appendix. The curves run from (a2, γb2, γ

2c2) = (0, 0, 0), the limiting point forlow input activity (〈r〉 = ∞), to (a2, γb2, γ

2c2) = (1, 0, 0), the limit of high-frequency stimulation (〈r〉 = 0). For γsufficiently small, the curves lie completely within the region of convergence. For larger γ, they are in the region ofdivergence except near the point (a2, γb2, γ

2c2) = (0, 0, 0). For 〈r/τ〉 ¿ 1, which is the realistic regime for corticalneurons, the eigenvalues ν2/3 are complex conjugate and we get

ν1 = 1−(

2− eγ

2

)⟨ r

τ

⟩

+O

(⟨ r

τ

⟩3/2)

, (48)

∣∣ν2/3∣∣ = 1−

(

2 +eγ

4

)⟨ r

τ

⟩

+O

(⟨ r

τ

⟩3/2)

. (49)

It follows that for 〈r/τ〉 ¿ 1, it is necessary and sufficient for the second moments to converge that γ < 4/e. In fact,Fig. 4 shows that at least for γ ≤ 1.2 the second moments converge for all 0 < 〈r/τ〉 <∞, corresponding to the entirecurves running between (a2, γb2, γ

2c2) = (0, 0, 0) and (a2, γb2, γ2c2) = (1, 0, 0) in the parameter space.

As shown in the Appendix, the condition γ < 4/e is for Poisson statistics of the times tj equivalent to 4a1 > γb1for all a1 ∈ (0, 1). In the following, we will assume this condition to hold. The system is thus always in the regime ofdamped oscillations of 〈V (t)〉; cf. Fig. 1.

After some lengthy but straightforward algebra, we find for the asymptotic variance of x, and hence of V (t),

limt→∞

var[V (t)] = var∞(x) :=⟨x2⟩

∞− 〈x〉2∞ =

⟨s2⟩ρ1 − 〈s〉2 ρ2 , (50)

with coefficients

ρ1 =γ2(b22 + 2c2 − 2a2c2

)

2(1− a2)3+ 4γ (1− a2) b2 + γ2b2

2 − 2γ2 (1 + a2) c2, (51)

ρ2 =γ2b1

2

[

(1− a1)2+ γb1

]2 (52)

− 2γ2[a1 (1− a1)

(b22 + 2c2 − 2a2c2

)+ b1 (b2 − a2b2 − 2γc2)

]

[

(1− a1)2+ γb1

] [

2(1− a2)3+ 4γ (1− a2) b2 + γ2b2

2 − 2γ2 (1 + a2) c2

] .

For Poisson statistics of the stimulus times tj , the coefficients ρ1/2 simplify to

ρ1 =(eγ)2

4eγ − (eγ)2 + 4 〈r/τ〉 > 0 , (53)

ρ2 =(eγ)3 (eγ + 2 〈r/τ〉)

[4eγ − (eγ)2 + 4 〈r/τ〉] (eγ + 〈r/τ〉)2> 0 ; (54)

cf. the Appendix.

D. Stationary states, fluctuations, and noise-driven oscillations

We have seen in the two previous sections that there is a region of stimulus parameters where the mean and varianceof V (t) converge to finite values. Averages do not tell us, however, what individual trajectories V (t) look like. In thissection we want to gain insight into the temporal pattern of individual trajectories.

9

00.2

0.40.6 0.8 1

01

23

0

2

4

6

8

00.2

0.40.6 0.8 1

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

0

123

0

2

4

6

8

0 0.2 0.4 0.6 0.8 1

00.20.40.60.8

1

0 1 2 3

0

2

4

6

8

00.20.40.60.8

1

0

2

4

6

8

PSfrag replacements

a2

a2

a2

γb2

γb2

γb2

γ2c2

γ2c2

γ2c2

γ = 0.2

γ = 3.7

FIG. 4: Three different views of the two surfacesdefined in Eq. (47) in the space of stimulus param-eters a2, b2, and c2. The region of parameters thatresult in convergence of the second moment of themembrane potential is the space between the twosurfaces that includes the a2 axis. Parameters forPoissonian stimulus times lie on the thick curvesfor γ = 0.2, 0.7, . . . , 3.7. The graphs show thatconvergence is ensured for all Poissonian stimuli,if γ ≤ 1.2.

10

20 40 60 20 40 60 20 40 60

5 10 15 5 10 15 5 10 15

2 4 6 2 4 6 2 4 6

PSfrag replacements

t/τ

V

0

〈s〉

∆s = 0.1 〈s〉 ∆s = 〈s〉 ∆s = 10 〈s〉

〈r/τ〉 = 0.001

〈r/τ〉 = 0.01

〈r/τ〉 = 0.1

FIG. 5: Typical trajectories of the membrane potential V (t), simulated for various stimulus conditions as indicated by the rowand column labels of the array of graphs. The unit of time is the rise time τ of PSPs [cf. Eq. (4)]. Synaptic reversal potentialsare uniformly distributed in the intervals [〈s〉 −∆s, 〈s〉+∆s]. The membrane potentials V = 0 (resting potential) and V = 〈s〉are indicated in each graph by the solid and dashed lines, respectively. The graphs show the transitions between stationary,fluctuating, and oscillatory dynamics of V (t) as discussed in the main text. The PSP-amplitude factor γ = 0.1 for all graphs.

Let us first deal with the short-time behavior of individual trajectories (xj , yj). We ask what they look like for thefirst few j, that is, the first few synaptic inputs. The variances

varj(x) :=⟨x2⟩

j− 〈x〉2j , varj(y) :=

⟨y2⟩

j− 〈y〉2j (55)

are zero initially. They increase to finite values no faster than the fastest-growing linear combination of secondmoments, i.e., like e−j/Q with

Q = −1/

ln

(

mini=1,2,3

|νi|)

. (56)

We have to compare Q to the period P of the oscillation of the mean values (〈x〉j , 〈y〉j) in order to see whether thisoscillation shows up in individual realizations (xj , yj). From Eqs. (34), (48), and (49) we obtain

P

Q=

(8 + eγ)π

2(eγ)1/2

⟨ r

τ

⟩1/2

+O(⟨ r

τ

⟩)

. (57)

Thus for 〈r/τ〉 sufficiently small, we get P/Q ¿ 1 and the oscillation of the means 〈x〉j , 〈y〉j is fast as comparedto the growth time of the fluctuations varj(x), varj(y) around the means. Individual realizations (xj , yj) are thenwell described by their means for several periods of the oscillation. Put differently, an oscillation with a mean periodgiven by Eq. (38) then shows up in individual realizations V (t). With longer interstimulus times 〈r/τ〉, fluctuationsincreasingly interfere with the oscillation. The transition from an oscillation-dominated to a fluctuation-dominateddynamics of V (t) is depicted in Fig. 5.

It remains to establish the long-time behavior of trajectories (xj , yj). The dynamics (33) of their mean valuespirals into the point (〈x〉∞, 〈y〉∞). Without damping of the oscillation, the trajectories would lie on orbits definedby q(x− 〈x〉∞, y − 〈y〉∞) = const, with the quadratic form

q(ξ, η) :=⟨(ξ, η), (KL)†KL(ξ, η)

⟩=

4γa21ξ2

b1− 4γa1ξη + 4a1η

2 . (58)

11

To estimate the true degree of damping of individual trajectories (xj , yj), we calculate the mean asymptotic ratio〈q/q0〉∞ with the initial value q0 := q(〈x〉∞, 〈y〉∞) of the quadratic form q. From Eq. (40) we obtain the threesecond moments 〈x2〉∞, 〈xy〉∞, and 〈y2〉∞ which are needed for the calculation of 〈q〉∞. After some lengthy butstraightforward algebra, we find

⟨q

q0

⟩

∞

=

⟨s2⟩

〈s〉2ρ1 − ρ2 , (59)

where the coefficients are for Poisson statistics of synaptic input times tj ,

ρ1 =2 (eγ + 〈r/τ〉)2

4eγ − (eγ)2 + 4 〈r/τ〉 > 0 , (60)

ρ2 =2eγ(eγ + 2 〈r/τ〉)

4eγ − (eγ)2 + 4 〈r/τ〉 > 0 ; (61)

cf. the Appendix. For var(s) = 〈s2〉 − 〈s〉2 = 0, it follows that

⟨q

q0

⟩

∞

= ρ1 − ρ2 =2 〈r/τ〉2

4eγ − (eγ)2 + 4 〈r/τ〉 <1

2

⟨ r

τ

⟩

¿ 1 . (62)

Hence there is strong damping, and individual trajectories (xj , yj) converge close to the steady mean state, if synapticcurrents have all the same reversal potential. On the other hand, for var(s)/〈s〉2 À 1, hence 〈s2〉/〈s〉2 À 1, weget 〈q/q0〉∞ À 1 and there is no damping of individual trajectories (xj , yj). Since the dynamics is a temporallyhomogeneous Markov chain, at any time we then find qualitatively the same situation as at the start of the process.Thus there is no qualitative change in the trajectories (xj , yj) on a long time scale, and the pattern of evolution, randomfluctuations or oscillations, that dominates initially (see above) will also prevail at all times. Figure 5 summarizes thetypes of dynamics of V (t), illustrating our results on short- and long-time behavior by computer simulations.

With synaptic reversal potentials sj having a high variance, we have seen individual trajectories V (t) to oscillate orfluctuate persistently around the value limt→∞〈V (t)〉 = 〈x〉∞. It is interesting to compare the mean of the intervals∆ = t− t′ between successive times t > t′ defined by

V (t) = V (t′) = 〈x〉∞ ,d

dtV (t) > 0 ,

d

dtV (t′) > 0 , (63)

the mean “jitter period”, with the mean oscillation period 〈T 〉 [cf. Eq. (38)] of 〈V (t)〉. I have measured jitter periodsin computer simulations of V (t). As can be seen in Fig. 6, the match between the two periods is perfect for small〈r/τ〉, that is, in the regime where oscillations are rather regular. For increasing interstimulus times 〈r/τ〉, when therandom-walk component of membrane dynamics grows stronger (cf. Fig. 5), the mean jitter period drops below themean oscillation period, indicating that fluctuations cause V (t) to jitter around its asymptotic mean value faster thanthe oscillatory component of the dynamics alone.

E. Delays

The conduction of synaptic currents in neuronal dendrites leads to delays relative to the time of the synaptic input.Let us assume here that we can assign a delay di > 0 to a PSP initiated at time ti, such that at time ti + di theresponse is spread out across the whole neuron. Of course, such a delay does not properly describe gradual PSPpropagation. In a sense, it is the opposite extreme of the instantaneous PSP propagation that we have considered sofar. The dynamics of the membrane potential with such delayed PSPs is given by

V (t) =

∞∑

i=1

Λ(γi, si, ti; t− di) ; (64)

cf. Eq. (7). A reformulation as a Markov chain as in Sec. IIIA is now not possible. This fact calls for a reconsiderationof our previous results. Here I am concerned with proving structural stability of the dynamics analyzed above withrespect to small delay perturbations. To this end, we may extend the previous dynamics to incorporate first-orderdelay effects. The issue of delays is covered in detail in [15] for a slightly more general class of dynamical system. Inthis paper, I only sketch the way to proceed.

12

0.02 0.04 0.06 0.08 0.1

1

2

3

4

PSfrag replacements

〈r/τ〉

〈T/τ〉

〈∆/τ〉

FIG. 6: Comparison of the mean oscillation period 〈T 〉 of the membrane potential as given by Eq. (38) (solid line; cf. Fig. 3)with the mean jitter period 〈∆〉 defined in Eq. (63) as observed in computer simulations (box symbols; bars indicate standarderrors). The unit of time is the rise time τ of PSPs [cf. Eq. (4)]. The match between the two periods is perfect for small〈r/τ〉, when oscillations are rather regular. As oscillations are increasingly degraded by fluctuations for larger 〈r/τ〉 (cf. Fig. 5),the mean jitter period drops below the mean oscillation period. In the simulations, synaptic reversal potentials are uniformlydistributed in an interval with 〈s〉 = 0; the PSP-amplitude factor γ = 0.1.

Expanding Eq. (64) to first order in the delays di/τ , we have to note that Λ(γi, si, ti; t) is not differentiable at t = ti;cf. Eq. (4). We can take advantage of the fact, however, that di > 0 and write

Λ(γi, si, ti; t− di) (65)

= Λ(γi, si, ti; t) + di limd→0+

Λ(γi, si, ti; t)− Λ(γi, si, ti; t− d)

d+O(d2i )

= Λ(γi, si, ti; t) +diτ

(

1− t

τ

)

e1−t/τ Θ(t− ti) +O

[(diτ

)2]

,

that is, we take the derivative of Λ(γi, si, ti; t) from lower values of t. Equation (65) is substituted into the dynamicEq. (64) and only terms up to first order in di/τ are kept. As before, we use γ ≡ γi to obtain a model with a minimalset of variables. We can now transform to new dynamic variables xj := V (tj), yj , and zj that obey the stochasticiteration

xjyjzj

= R′(rj−1) ◦ S ′(sj−1, dj−1)

xj−1yj−1zj−1

, x1 = y1 = z1 = 0 , (66)

S ′(s, d) :

xyz

7→

xy + γ(s− x)

(1 + d

τ

)

z + γ(s− x) dτ

, (67)

R′(r) :

xyz

7→

(x+ ey r

τ − ez)e−r/τ

ye−r/τ

ze−r/τ

. (68)

The dimension of the stochastic dynamic map is increased by one compared to the case without delays; cf. Eq. (12).Treating Eq. (66) analogous to Eq. (12), we can derive dynamic maps for the moments of x, y, and z. These will haveaccordingly higher dimensions than those for the moments of x and y without delays. This underlines the necessityto check the structural stability of the dynamics derived previously.

It can be shown [15] that the dynamics for the first and second moments of x and y is stable with respect to smalldelay perturbations, provided that

γb2 < 2(a2 + 1)2 . (69)

In general, this is a condition for convergence in the delayed system that is additional to those derived for theundelayed system. For Poisson statistics of synaptic input times, however, we know that a2 and b2 lie on the parabola

13

b2 = ea2(1− a2); see the Appendix. Together with the condition γ < 4/e derived in Sec. III C for the convergence ofthe variance of V (t), this implies condition (69).

By continuity of eigenvalues and asymptotic values in the delays within the extended model, it follows that forsmall delays there is only a small quantitative and no qualitative change in membrane dynamics. All that has beenconcluded on patterns of the dynamics hence remains true for small delays. Moreover, it can be shown that smalldelays decrease the asymptotic attraction of 〈V (t)〉 to the mean synaptic reversal potential 〈s〉 and increase the meanperiod 〈T/τ〉 of membrane oscillations [15].

IV. SUMMARY AND DISCUSSION

In this paper, I have analyzed the subthreshold dynamics of the neural membrane potential driven by stochasticsynaptic input of stationary statistics. Conditions on the input statistics for stability of the dynamics have beenderived. Regimes of input statistics for stationary, fluctuating, and oscillatory dynamics have been identified. For thecase of Poissonian stimulus times, that is, temporal noise, it has turned out that persistent oscillations can developwith a mean period that depends nontrivially on the mean interstimulus time. In particular, noise-driven oscillationsoccur in the absence of any pace-making mechanism in the stimulus, in the intrinsic neural dynamics, or in a recurrentneural network.

What does it mean for a real neuron, if its membrane potential is “unstable” under stimulation by the network’ssynaptic input? As the analysis has shown, instability of the first or second moments implies excursions of V (t) withgrowing positive and negative amplitudes. After some stochastic period of time, therefore, the membrane potentialwill certainly cross the threshold for firing. The neuron will then be set to a post-spike potential that depends insome way on the stimulus history and the process will resume.

I have neglected many effects in the modeling for the sake of analytical feasibility. Most notably, PSPs have beengiven a shape that does not properly reflect conduction in neuronal dendrites. One shortcoming is a lack of variabilityof PSP shape; see, however, [7, 9, 19]. Another is that real neuronal membranes contain ionic conductances which arevoltage-gated [14]. Their effect is to modify the shape of PSPs in a voltage-dependent manner as they are propagatedalong a dendrite; see, e.g., [1]. Moreover, with voltage-gated channels, PSPs do not simply add up but interactnonlinearly. The conclusions drawn in the present paper, therefore, can only be on qualitative system behavior andshould not be understood quantitatively.

In the analyzed model, there is no representation of the spatial dimensions of a neuron. For a neuron where spatialconduction times are significant, the present results suggest that spatiotemporal waves of membrane potential developin the regime of noise-driven oscillations. For the generation of action potentials, however, all that matters is thepotential at the cell’s soma.

A. Oscillations in stochastic systems

It is a common example in textbooks on stochastic dynamical systems to calculate stationary densities for adamped harmonic oscillator subject to an external stochastic force; see, e.g., [17]. If the damped oscillator is in theperiodic regime, the intrinsic oscillations are sustained by the stochastic force. In the context of biological systems,stochastically sustained oscillations have been analyzed, somewhat heuristically, for the population dynamics of anepidemic model [2]. This system is autonomous and an intrinsic oscillator. The stochastic nature of the dynamicsprevents asymptotic convergence to a steady state.

It is thus a known generic property of periodic relaxation systems to exhibit oscillations at their intrinsic frequency,sustained by some stochastic influence. The dynamics analyzed in this paper, however, represents a different typeof phenomenon. The system studied is not an intrinsic oscillator but exhibits oscillations at a mean period that is,up to a temporal scale, determined by the stochastic drive alone. The system can be formally viewed as a controlloop where a sequence of brief signals (the synaptic reversal potentials sj) controls via a slow response (the PSPs) adynamic variable [the membrane potential V (t)]. The theme of the control loop is fully developed in [15, 16].

B. Oscillations in neural systems

Oscillations of membrane potential and spiking activity are quite ordinary in the neural systems of the brain. Theyarise under various conditions, with varying degree of correlation between neurons, and in a wide range of frequencies.Their functional implications may be equally various and are much debated today.

14

1000 1500 1000 1500 1000 1500

PSfrag replacements

t/τ t/τ t/τ

V V V

0

sdep

shyp

FIG. 7: Demonstration of hyperpolarization-induced activity. The three graphs show long-time simulations of the membranepotential V (t). Transients at the start of the simulation are cut off. The unit of time is the rise time τ of PSPs [cf. Eq. (4)].Depolarizing synaptic input is applied with mean interstimulus time 〈r/τ〉 = 0.1 and reversal potential sdep > 0 as indicatedby the upper dashed line in each graph. There is no hyperpolarizing synaptic input for the left graph; for the central graphthere is hyperpolarization with 〈r/τ〉 = 0.5; for the right graph with 〈r/τ〉 = 0.1. The hyperpolarizing reversal potential isshyp = −sdep < 0 as indicated by the lower dashed line in each graph. If the threshold for spike generation is assumed closeto sdep, there will be no spikes for the case without hyperpolarizing input (left), a few spikes for weak hyperpolarizing input(center), and again no spikes for equal hyperpolarizing and depolarizing input (right). The PSP-amplitude factor γ = 0.1 forall graphs.

Explanations of oscillations have been basically of two kinds. One is in terms of intrinsic oscillator cells that actas a pacemaker for rhythmic activity in the network [12, 29]. The other makes reference to the fact that recurrentneural networks have a natural tendency to produce rhythmic and synchronized activity [5, 11].

Some of neural oscillations are most probably generated by the intrinsic neural dynamics of ion channels. Othersare propagated by synaptic potentials and are of less certain origin. A prominent example of the latter kind arecortical oscillations in the gamma frequency band (roughly 20–90 Hz). Cortical gamma oscillations are mostly evokedby a sensory stimulus. Thus, spontaneous activity in the visual cortex of awake cats and primates is rarely oscillatory,whereas visual stimuli of increasing speed of motion produce subthreshold and suprathreshold oscillations of increasingfrequency [4, 6, 10, 13, 20, 21].

The results presented here suggest that oscillations of the neural membrane potential can arise from the network’sbackground activity. Let us assume that a stimulus evokes responses in neurons of a coupled system at a rate thatincreases with stimulus speed, because more neurons in the network are stimulated per time at higher speeds [32].The observed dependence of oscillations on a stimulus is then predicted by Sec. IIID, the relation between oscillationperiod and stimulus speed by Eq. (38). Note that the conditions 〈r/τ〉 ¿ 1 and var(s)/〈s〉2 À 1 for the developmentof noise-driven oscillations are probably fulfilled under external stimulation of a network of cortical neurons, eachreceiving roughly 10000 synapses of both an excitatory and inhibitory kind [8]. The degree of correlation betweenneurons that is to be expected from noise-driven oscillations increases with the extent to which they share commoninput from the network’s background activity. Correlations should, therefore, decrease with distance between neurons,in agreement with what is generally observed. In a network of spike-exchanging neurons, however, correlations caneven arise between neurons that do not share any input.

The present analysis draws attention to a phenomenon, noise-driven oscillations, that should be very common inneural systems and may be the cause of some of the observed membrane-potential oscillations.

C. Hyperpolarization-induced activity

There is an interesting consequence of the analytical results. It is, at first sight, somewhat counterintuitive. Considera neuron that receives depolarizing synaptic input at a fixed average rate. Let us assume that at this level ofdepolarization the membrane potential remains mostly below the threshold for spike generation. Now, if we addsome hyperpolarizing synaptic input, it turns out that the neuron may actually start spiking. Further increase of thehyperpolarizing input rate eventually shuts neural activity off. This scenario is demonstrated in computer simulationsshown in Fig. 7.

The effect seems to be at odds with the usual notion of hyperpolarizing synapses to inhibit neural activity rather thanpromote it. Exceptions have only been reported for cases where a hyperpolarization-activated current repolarizes thecell, giving rise to a rebound burst of action potentials; see, e.g., [18, 22]. The effect demonstrated here is of a differentnature. It results from an increase of membrane fluctuations with the addition of hyperpolarizing synaptic input; cf.Eq. (50). For a range of hyperpolarizing input rates, increased fluctuations are likely to spontaneously overcome theassociated drop in mean membrane potential; cf. Eq. (29). The result is fluctuation-driven spike generation.

15

The phenomenon of hyperpolarization-induced activity offers a subtle way in which neural spiking may be controlled.Whether it is actually used in the brain is unexplored today.

APPENDIX

It is reasonable to assume the times tj at which synaptic inputs are received by a cortical neuron from other corticalneurons to obey Poisson statistics. For the density u of interstimulus times r this means

u(r) =e−r/〈r〉

〈r〉 . (A.1)

In order to transform the stimulus parameters ai, bi, ci introduced in Eqs. (26) and (41), and to reveal dependencesbetween them, we calculate the mean values

⟨( r

τ

)k

e−r/τ

⟩

=

(

− ∂

∂α

)k ⟨

e−αr/τ⟩∣∣∣∣∣α=1

=

(

− ∂

∂α

)k ∫ ∞

0

dr u(r) e−αr/τ

∣∣∣∣∣α=1

=

(

− ∂

∂α

)k1

1 + α 〈r/τ〉

∣∣∣∣∣α=1

. (A.2)

Hence the stimulus parameters turn out to be

a1 =1

1 + 〈r/τ〉 , b1 =e 〈r/τ〉

(1 + 〈r/τ〉)2, (A.3)

a2 =1

1 + 2 〈r/τ〉 , b2 =2e 〈r/τ〉

(1 + 2 〈r/τ〉)2, c2 =

2e2 〈r/τ〉2

(1 + 2 〈r/τ〉)3.

Dependences between these parameters are now explicit. In particular, we have

bi = e ai (1− ai) , i = 1, 2 . (A.4)

In Sec. III C, we have established that for Poisson statistics and small 〈r/τ〉 the necessary and sufficient conditionfor the second moment of V (t) to converge is γ < 4/e. Multiplying Eq. (A.4) by γ, we see that this bound implies

γbi < 4 ai (1− ai) < 4 ai for ai ∈ (0, 1), i = 1, 2. (A.5)

Conversely, γbi < 4ai for all ai ∈ (0, 1) together with Eq. (A.4) implies γ < 4/e.

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[31] Conduction times of ionotropic synapses are roughly 1 ms, while the membrane time constant is usually several 10 ms [14].The other type of synapse, the metabotropic synapse, modulates the membrane potential on a much longer time scale ofseconds [14] and is not considered here.

[32] Different individual neurons respond best at different stimulus speeds, a property called speed tuning. For the completecoupled system, however, we may neglect effects of speed tuning.